Integrate sin^2(x)

sin^2(x) is not one of the functions whose antiderivative is commonly known to us. When we see trigonometric functions our best bet is usually to think of some trigonometric identity we know. In this case we want an identity which will relate sin^2(x) to a function we can integrate. A little thought tells us that the cosine double angle formula helps. This is cos(2x)=1-2sin^2(x). Rearrange to make sin^2(x) the subject of the formula: sin^2(x)=(1-cos(2x))/2. Now we can integrate:Integral(sin^2(x))=Integral((1-cos(2x))/2)=Integral(1/2-cos(2x)/2). Now integrate term by term (as integration is linear). The integral of 1/2 with respect to x is x/2. The integral of cos(2x)/2 is sin(2x)/4 (check: derivative of sin(2x) is 2cos(2x) so derivative of sin(2x)/4 is indeed cos(2x)/2). So our answer is thus Integral(sin^2(x))=x/2-sin(2x)/4+c, where c is a constant of integration.